Mathematical theory of scattering resonances

649 Pages·2019·32.503 MB·English

by  Dyatlov, Semyon, Zworski, Maciej

GRADUATE STUDIES 200
IN MATHEMATICS
Mathematical
(cid:56)(cid:76)(cid:73)(cid:83)(cid:86)(cid:93)(cid:3)(cid:83)(cid:74)(cid:3)
Scattering
Resonances
Semyon Dyatlov
Maciej Zworski
Mathematical
Theory of
Scattering
Resonances

GRADUATE STUDIES 200
IN MATHEMATICS
Mathematical
Theory of
Scattering
Resonances
Semyon Dyatlov
Maciej Zworski
EDITORIAL COMMITTEE
Daniel S. Freed (Chair)
Bjorn Poonen
Gigliola Staffilani
Jeff A. Viaclovsky
2010 Mathematics Subject Classification. Primary 58J50, 35P25, 34L25, 35P20,35S05,
81U20, 81Q12, 81Q20.
For additional informationand updates on this book, visit
www.ams.org/bookpages/gsm-200
Library of Congress Cataloging-in-Publication Data
Names: Dyatlov,Semyon,1987-author. |Zworski,Maciej,author.
Title: Mathematicaltheoryofscatteringresonances/SemyonDyatlov,MaciejZworksi.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2019]|Series: Gradu-
atestudiesinmathematics;volume200|Includesbibliographicalreferencesandindex.
Identifiers: LCCN2019006281|ISBN9781470443665(alk. paper)
Subjects: LCSH: Scattering (Mathematics)–Problems, exercises, etc. | Oscillations–Problems,
exercises, etc. | Frequencies of oscillating systems–Problems, exercises, etc. | Mathematical
physics–Problems, exercises, etc. | AMS: Partial differential equations – Spectral theory and
eigenvalueproblems–Scatteringtheory. msc|Ordinarydifferentialequations–Ordinarydif-
ferentialoperators– Scattering theory, inverse scattering. msc|Partialdifferential equations
–Spectraltheoryandeigenvalueproblems–Asymptoticdistributionofeigenvaluesandeigen-
functions. msc|Quantumtheory–Scatteringtheory–S-matrixtheory,etc.. msc|Quantum
theory – General mathematicaltopics and methods in quantum theory – Non-selfadjoint op-
erator theory in quantum theory. msc | Quantum theory – General mathematicaltopics and
methodsinquantumtheory–Semiclassicaltechniques,includingWKBandMaslovmethods.
msc
Classification: LCCQA329.D932019|DDC515/.946–dc23
LCrecordavailableathttps://lccn.loc.gov/2019006281
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Contents
Preface ix
Chapter 1. Introduction 1
§1.1. Resonances in scattering theory 1
§1.2. Semiclassical study of resonances 7
§1.3. Some examples 8
§1.4. Overview 13
Part 1. POTENTIAL SCATTERING
Chapter 2. Scattering resonances in dimension one 21
§2.1. Outgoing and incoming solutions 22
§2.2. Meromorphic continuation 26
§2.3. Expansions of scattered waves 39
§2.4. Scattering matrix in dimension one 45
§2.5. Asymptotics for the counting function 52
§2.6. Trace and Breit–Wigner formulas 59
§2.7. Complex scaling in one dimension 70
§2.8. Semiclassical study of resonances 82
§2.9. Notes 91
§2.10. Exercises 92
Chapter 3. Scattering resonances in odd dimensions 95
§3.1. Free resolvent in odd dimensions 96
§3.2. Meromorphic continuation 108
v
vi Contents
§3.3. Resolvent at zero energy 116
§3.4. Upper bounds on the number of resonances 125
§3.5. Complex-valued potentials with no resonances 129
§3.6. Outgoing solutions and Rellich’s theorem 131
§3.7. The scattering matrix 143
§3.8. More on distorted plane waves 155
§3.9. The Birman–Kre˘ın trace formula 159
§3.10. The Melrose trace formula 177
§3.11. Scattering asymptotics 187
§3.12. Existence of resonances for real potentials 205
§3.13. Notes 207
§3.14. Exercises 210
Part 2. GEOMETRIC SCATTERING
Chapter 4. Black box scattering in Rn 217
§4.1. General assumptions 218
§4.2. Meromorphic continuation 223
§4.3. Upper bounds on the number of resonances 235
§4.4. Plane waves and the scattering matrix 250
§4.5. Complex scaling 268
§4.6. Singularities and resonance-free regions 289
§4.7. Notes 300
§4.8. Exercises 303
Chapter 5. Scattering on hyperbolic manifolds 305
§5.1. Asymptotically hyperbolic manifolds 307
§5.2. A motivating example 314
§5.3. The modified Laplacian 317
§5.4. Phase space dynamics 323
§5.5. Propagation estimates 332
§5.6. Meromorphic continuation 341
§5.7. Applications to general relativity 351
§5.8. Notes 362
§5.9. Exercises 364
Contents vii
Part 3. RESONANCES IN THE SEMICLASSICAL LIMIT
Chapter 6. Resonance-free regions 371
§6.1. Geometry of trapping 373
§6.2. Resonances in strips 380
§6.3. Normally hyperbolic trapping 392
§6.4. Logarithmic resonance-free regions 403
§6.5. Lower bounds on resonance widths 408
§6.6. Notes 418
§6.7. Exercises 421
Chapter 7. Resonances and trapping 425
§7.1. Lower bounds on the resolvent 426
§7.2. Semiclassical growth estimates 432
§7.3. From quasimodes to resonances 437
§7.4. The Sjo¨strand trace formula 446
§7.5. Resonance expansions for strong trapping 456
§7.6. Notes 467
§7.7. Exercises 468
Part 4. APPENDICES
Appendix A. Notation 475
§A.1. Basic notation 475
§A.2. Functions 476
§A.3. Spaces of functions 477
§A.4. Operators 477
§A.5. Estimates 478
§A.6. Tempered distributions 479
§A.7. Distributions on manifolds and Schwartz kernels 480
Appendix B. Spectral theory 483
§B.1. Spectral theory of self-adjoint operators 483
§B.2. Functional calculus 488
§B.3. Singular values 489
§B.4. The trace class 492
§B.5. Weyl inequalities and Fredholm determinants 497
§B.6. Lidski˘ı’s theorem 504
viii Contents
§B.7. Notes 506
§B.8. Exercises 506
Appendix C. Fredholm theory 507
§C.1. Grushin problems 507
§C.2. Fredholm operators 509
§C.3. Meromorphic continuation of operators 513
§C.4. Gohberg–Sigal theory 516
§C.5. Notes 524
§C.6. Exercises 524
Appendix D. Complex analysis 527
§D.1. General facts 527
§D.2. Entire functions 531
Appendix E. Semiclassical analysis 535
§E.1. Pseudodifferential operators 536
§E.2. Wavefront sets and ellipticity 556
§E.3. Semiclassical defect measures 566
§E.4. Propagation estimates 569
§E.5. Hyperbolic estimates 586
§E.6. Notes 603
§E.7. Exercises 604
Bibliography 613
Index 631
PREFACE
Mathematicians are Frenchmen of sorts: whatever one says to
them they translate into their own language and then it becomes
something entirely different.
Johann Wolfgang von Goethe, Maximen und Reflexionen, 1840
Thepurposeofthisbookistoprovideabroadintroductiontothetheory
of scattering resonances.
Scattering resonances appear in many branches of mathematics, physics
and engineering. They generalize eigenvalues or bound states for systems in
which energy can scatter to infinity. A typical state has then a rate of oscil-
lation (just as a bound state does) and a rate of decay. Although the notion
is intrinsically dynamical, an elegant mathematical formulation comes from
considering meromorphic continuations of Green’s functions or scattering
matrices. The poles of these meromorphic continuations capture the phys-
ical information by identifying the rate of oscillations with the real part of
a pole and the rate of decay with its imaginary part. The resonant state,
which is the corresponding wave function, then appears in the residue of the
meromorphically continued operator. An example from pure mathematics
is given by the zeros of the Riemann zeta function: they are, essentially, the
resonances of the Laplacian on the modular surface. The Riemann hypoth-
esis then states that the decay rates for the modular surface are all either
0 or 1. A standard example from physics is given by shape resonances cre-
4
ated when the interaction region is separated from free space by a potential
barrier. The decay rate is then exponentially small in a way depending on
the width of the barrier.
ix